02 binary numbers

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Instructor: Alexander Stoytchev

http://www.ece.iastate.edu/~alexs/classes/

CprE 281:i!ital "o!ic


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#inary $u%&ers

CprE 281: Digital Logic

Iowa State University, Ames, IACopyright Ale!an"er Stoytchev


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Ad%inistrative Stu''

(his is the o''icial class we& pa!e:

http://www.ece.iastate.edu/~alexs/classes/2)1*+Sprin!+281/

I' you %issed the 'irst lecture, the sylla&us and other class

%aterials are posted there.

http://www.ece.iastate.edu/~alexs/classes/2014_Spring_281/http://www.ece.iastate.edu/~alexs/classes/2014_Spring_281/http://www.ece.iastate.edu/~alexs/classes/2014_Spring_281/


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• -1 is out

• It is due on ednesday e& 22 0 *p%.

• Su&%it it on paper &e'ore the start o' the lecture


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(he la&s and recitations start next wee:

•Section : (uesdays, 12:1) 3:))p% 4Coover -all, roo% 2)5)6

•Section 7: (uesdays, 3:1) :))p% 4Coover -all, roo% 2)5)6

•Section 9: ednesdays, 12:1) 3:))p% 4Coover -all, roo% 2)5)6

•Section : ednesdays, :1) ;:))p% 4Coover -all, roo% 2)5)6

•(he la& schedule is posted on the class we& pa!e


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"a&s $ext ee

Figure 1.5 in the textbook: An FPGA board.


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"a&s $ext ee

•


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• $o class next >onday 4?niversity -oliday6


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(he eci%al Syste%

[https://reader015.{domain}/reader015/html5/0716/5b4bf4a9


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hat nu%&er syste% is this one@

[http://freedomhygiene.com/wp-content/themes/branfordmagazine/images/backgrounds/Hands_141756.jpg]


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(he #inary Syste%

[ http://divaprojections.blogspot.com/2011/11/alien.html]


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$u%&er Syste%s


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$u%&er Syste%s

n-th digit

(most significant)

0-th digit

(least significant)


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(he eci%al Syste%


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(he eci%al Syste%


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Another ay to "oo at (his

5 2 4


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5 2 4

102 101 100


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5 2 4

102 101 100

boxes

labels

Each box can contain only one digit and has only one label. From right

to left, the labels are increasing powers of the base, starting from 0.


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#ase


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#ase

base power


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#ase

most significant

digit

least significant

digit

base power


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#ase


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#inary $u%&ers 4#ase 26


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most significant bit least significant bit

base power


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Another Exa%ple


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hat is the value o' this &inary nu%&er@

• ) ) 1 ) 1 1 ) )

• ) ) 1 ) 1 1 ) )

• )B2A )B2 1B25 )B2* 1B23 1B22 )B21 )B2)

• )B128 )B* 1B32 )B1 1B8 1B* )B2 )B1

• )B128 )B* 1B32 )B1 1B8 1B* )B2 )B1

• 32 8 * D ** 4in deci%al6


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0 0 1 0 1 1 0 02021222324252627


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So%e (er%inolo!y

• A &inary di!it is called a bit

• A !roup o' ei!ht &its is called a &yte

• ne &it can represent only two possi&le states,which are denoted with 1 and )


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Felationship #etween a #yte and a #it

1 0 1 0 1 1 1 0


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1 0 1 0 1 1 1 0

1 bit


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1 0 1 0 1 1 1 0

8 bits = 1 byte

1 bit


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#it


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#it


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hat is the %axi%u% nu%&er that can &estored in one &yte 48 &its6@

hat is the %axi%u% nu%&er that can &e


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hat is the %axi%u% nu%&er that can &estored in one &yte 48 &its6@

• 1 1 1 1 1 1 1 1

• 1 1 1 1 1 1 1 1

• 1B2A 1B2 1B25 1B2* 1B23 1B22 1B21 1B2)

• 1B128 1B* 1B32 1B1 1B8 1B* 1B2 1B1

• 128 * 32 1 8 * 2 1 D 255 4in deci%al6

• Another way is: 1B28 H 1 D 25 H 1 D 255

hat would happen i' we try to add 1 to the lar!est


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hat would happen i' we try to add 1 to the lar!est

nu%&er that can &e stored in one &yte 48 &its6@

1 1 1 1 1 1 1 1

1

1 ) ) ) ) ) ) ) )

) ) ) ) ) ) ) )


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Analo!y with car odo%eters


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Analo!y with car odo%eters

[http://www.hyperocity.com/volvo240/images/Volvo/odometerrepair/speedo999999.jpg]

eci%al to #inary Conversion


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4?sin! uessin!6



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4?sin! uessin!6


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Convertin! 'ro% eci%al to #inary


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Convertin! 'ro% eci%al to #inary


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[ Figure 1.6 in the textbook ]


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ctal Syste% 4#ase 86


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#inary to ctal Conversion


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1011100101112 = ?8


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1011100101112 = ?8

101 110 010 111


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1011100101112 = ?8

101 110 010 111

5 6 2 7


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1011100101112 = ?8

101 110 010 111

5 6 2 7

Thus, 1011100101112 =


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-exadeci%al Syste% 4#ase 16


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(he 1 -exadeci%al i!its

), 1, 2, 3, *, 5, , , 8, ;, A, #, C, , E,

1), 11, 12, 13, 1*, 15


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-exadeci%al to eci%al Conversion


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#inary to -exadeci%al Conversion


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#i t - d i l C i


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1011100101112 = ?16

#i t - d i l C i


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1011100101112 = ?16

1011 1001 0111

#i t - d i l C i


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1011100101112 = ?16

1011 1001 0111

B 9 7

#i t - d i l C i


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1011100101112 = ?16

1011 1001 0111

B 9 7

Thus, 1011100101112 = B9716

i l t - d i l C i


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eci%al to -exadeci%al Conversion

i l t - d i l C i


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eci%al to -exadeci%al Conversion

Si d i t li t d


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Si!ned inte!ers are %ore co%plicated

e will tal %ore a&out the% when we start withChapter 3 in a couple o' wees.

(he story with 'loats is even %ore co%plicated


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y p

IEEE 5*1;85 Standard

[http://en.wikipedia.org/wiki/IEEE_754]


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In the exa%ple shown a&ove, the sign is ero so sis 1, the exponent is 12* so e is J3, and thesi!ni'icand % is 1.)1 4in &inary, which is 1.25 indeci%al6. (he represented nu%&er is there'ore1.25 K 2J3, which is ).1525.

[http://en.wikipedia.org/wiki/IEEE_754]

n line IEEE 5* Con erters


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nline IEEE 5* Converters

• http://www.hsch%idt.net/loatApplet/IEEE5*.ht%l

• http://&a&&a!e.cs.Gc.edu/IEEE5*/eci%al.ht%l

• >ore a&out 'loatin! point nu%&ers in Chapter 3.

Storin! Characters

http://www.h-schmidt.net/FloatApplet/IEEE754.htmlhttp://babbage.cs.qc.edu/IEEE-754/Decimal.htmlhttp://babbage.cs.qc.edu/IEEE-754/Decimal.htmlhttp://www.h-schmidt.net/FloatApplet/IEEE754.html


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Storin! Characters

• (his reGuires so%e convention that %aps &inary

nu%&ers to characters.

• ASCII ta&le

• ?nicode

ASCII (a&le


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ASCII (a&le

Extended ASCII Codes


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Extended ASCII Codes

(he ?nicode Character Code


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(he ?nicode Character Code

• http://www.unicode.or!/charts/


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E!yptian

-iero!lyphs

http://www.unicode.org/charts/

Close up


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Close up

http://www.unicode.org/charts/


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Luestions@


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(-E E$

02 binary numbers

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